Abstract

We extend the response theory of optical forces to general electromagnetic systems which can be treated as multi-port systems with multiple mechanical degrees of freedom. We demonstrate a fundamental link between the scattering properties of an optical system to its ability to produce conservative or non-conservative optical forces. Through the exploration of two nontrivial two-port systems, including an analytical Fabry-Perot interferometer and a more complex particle-in-a-waveguide structure, we show perfect agreement between the response theory and numerical first-principle calculations. We show that new insights into the origins of optical forces from the response theory provide clear means of understanding conservative and non-conservative forces in a regime where traditional gradient force picture fails.

Figures (6)

Schematic of four main types of mechanically-variable optical systems: (a) a single-port system with a single degree of freedom (DOF); (b) a single-port system with n independent DOFs; (c) an M-port system with a single DOF; and (d) an M-port system with n independent DOFs.

Schematic of the field amplitudes in a lossless Fabry-Perot interferometer in vacuum. The incident wave enters the interferometer at normal angle from the left. The scattering matrix of the entire interferometer is taken at the two fixed reference planes. The cavity length l0 is the driving degrees of freedom and is marked in blue.

(a) Schematic of a high-index scattering cylinder (green) in a single-mode parallel-plate waveguide, with a width of a. The red arrow represents the incident direction. The blue and red regions are Ez fields with large positive and negative values. (b) Top view of the structure, where the blue and red regions are Ez fields. (c) Intensity distribution of the incident optical fields at power Pincident. After scattering, the transmitted field has a power of Pt and a phase of ϕt, while the reflected field has a power of Pr and a phase of ϕr. (d) The gradient force distribution derived from the intensity distribution. The scattering force follows the lateral distribution of the intensity plotted in (e).

The axial optical force (a) and the lateral force (b) are calculated from the Maxwell stress tensor (circles) for a silicon cylinder of radius 0.05a and from RTOF method (blue curve), in excellent agreement. The incident frequency is 1.36(c/2a) in both cases. Both Faxial (d) and Flateral (e and f) profiles exhibit large frequency dependence. Faxial only contains contributions from the reflection, while Flateral profile can be decomposed into the contributions from the reflection and from the transmission. All force profiles are plotted in a range slightly smaller than the full width of the waveguide, due to the finite size of the cylinder. The incident power is 1W/m in all cases.

Calculated axial forces and lateral forces as a function of the normalized frequency of the incidence and the location of the cylinder y. Three cases of cylinder radius, 0.05a, 0.075a and 0.125a, are compared. In all cases, only left half of the waveguide is shown, since with respect to the center, Faxial is symmetric and Flateral is antisymmetric. The color scale is chosen to compare the force amplitude and the maximum value (out-of-scale in the dark red region) of Flateral is calculated to be 14.8nN/a and 18.6nN/a for r = 0.075a and 0.125a respectively. The incident power is 1W/m in all cases. The dash lines in (a) and (b) indicate the frequencies discussed in Fig. 4. The gray regions in (e), (f), (h) and (i) are areas where the cylinder cannot move into because of its finite size.

The optical force fields (black arrows) of the two cases illustrated in Fig. 5. Colored areas represent the curl of the force field along the out-of-plane z direction. When the cylinder takes the trajectory indicated by the white contour on the left in panel (a), optical forces do work to the cylinder, because the total curl within the contour does not vanish. Only special trajectories inside which the positive curl region cancel the negative curl region, such as the white contour on the right, do not perform work onto the cylinder. (c) Schematic of the origin of nonconservative optical forces on a reflector. The grey rectangle represents a reflector, and the black arrows illustrate the directions and the amplitudes of the optical forces.